Proving Reciprocal Identities: (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x

[Schaus]

Homework Statement

(secx+1)/(sin2x) = (tanx)/2cosx-2cos²x)

Homework Equations

The Attempt at a Solution

Left Side
((1+cosx)/cosx)/2sinxcosx

((1+cosx)/cosx) x (1/2sinxcosx)
cancel the a cosx from both to get
(1/2sinxcosx)
This is all I could manage with left side so I tried right side
Right Side
(sinx/cosx)/2cosx-2cos²x)
I'm stuck here. I've been trying to find something to change the denominator of the Right Side but I can't think of anything that will work. If someone could let me know where I am going wrong it would be greatly appreciated!

 

[haruspex]

((1+cosx)/cosx) x (1/2sinxcosx)
cancel the a cosx from both to get
(1/2sinxcosx)

What happened to the 1+cos?

Rather than working each side separately, multiply out to get rid of all the denominators.

 

[Schaus]

I thought I could cancel a cosx, maybe I cannot. I tried to eliminate the denominator like you said. Here's what I got.
(secx+1)(2cosx-2cos²x) = (Sin2x)(tanx)

((1+cosx)/cosx)(2cosx-2cos²x)=(2sinxcosx)(sinx/cosx)
Expanding
((2cosx-2cos²x+2cos²x-2cos³x)/cosx) = 2
Moving the cosx under left side to the right side and simplifying
2cosx-2cos³x = 2cosx
Does this look right? And if so, where do I go from here?

 

[haruspex]

((2cosx-2cos2x+2cos2x-2cos3x)/cosx) = 2

Check the right hand side.

 

[Schaus]

Woops. I think I should have gotten 2sin²xcosx
2cosx-2cos³x = 2sin²xcosx
Does this look right?

 

[haruspex]

Woops. I think I should have gotten 2sin²xcosx
2cosx-2cos³x = 2sin²xcosx
Does this look right?

Yes. Keep simplifying.

 

[Schaus]

2cosx-2cos³x = 2sin²xcosx
2cosx(1-cos²x) = 2sin²xcosx
2cosx(sin²x) = 2sin²xcosx
2cosxsin²x = 2sin²xcosx
Does this work?

 

[haruspex]

2cosx-2cos³x = 2sin²xcosx
2cosx(1-cos²x) = 2sin²xcosx
2cosx(sin²x) = 2sin²xcosx
2cosxsin²x = 2sin²xcosx
Does this work?

Yes.
Of course, it is not strictly kosher to start with the thing to be proved and deduce a tautology. You need all the steps to be reversible. They are in this case, but it is cleaner to rewrite it in the more persuasive sequence: start with the tautology and deduce the thing to be proved.

 

[Schaus]

So I should start with 2sin²xcosx and work backwards? Thank you for all the help by the way.

 

[haruspex]

So I should start with 2sin²xcosx and work backwards?

Ideally, yes.

 

[Schaus]

Ok, I'll try it. I'm going to have to practice these quite a bit more I think.